Do things exist which aren't material - yes, of course. It really depends on what we mean by the word, "exist", which is probably too weak and non-specific a word to encompass what is we refer to when we say it. Frequently "existence" is applied both to material objects and to non-material entities: processes, flows, relationships, behaviors, evolution, and other dynamic aspects of complex (i.e., more than one object involved) systems. Obviously, emotions and feelings "exist". We humans (and probably other animals) clearly have inner lives. Love, hate, loyalty, treachery, and all the other human emotions "exist", but more in the sense of how relationships or processes exist. There are no emotional "atoms" that can be weighed and measured. But it is probable (neuroscience and evolutionary biology are very close to showing this) that the human emotions that give us reasons to keep on living are outgrowths of very physical processes in the brain. That doesn't make them any less important, just as knowing how a rainbow, opera, flower, or sonnet works doesn't make them any less beautiful. To those who would say, "if you can accept that love and loyality exist though you can't hold them in your hand, then why can you not also accept god?". The answer is that unlike love and loyalty, there is no evidence of a god, no reason to believe in a god, any more than there is a reason to believe in the infinity of other concepts that can be thought of which also don't exist. Knowing that a few immaterial "things" exist (emotion, change, the future, the past) is no argument for thinking every immaterial thing exists. Each requires its own rationale and reason for deserving our belief.
Material objects (including ourselves) are in the world. Those objects interact - they have relationships to each other and influence each other. Processes and phenomena occur, objects change internally and with respect to each other through time. Because of our cognitive apparatus, we create models of both the objects of the world, their relationships, and how both of these evolve through time. We see the universe work as it does, and build intricate mathematical and logical systems that correspond to the world.
"Embodied mind" theories hold that mathematical and logical thought is a natural outgrowth of the human cognitive apparatus which finds itself in our physical universe. For example, the abstract concept of number springs from the experience of living in a world where there are discrete objects that can be counted. I feel quite certain that if the universe did not have this feature (separate objects), that the theory of number would probably not have come about. Although it is not the dominant theory, I think that logical systems are a result of the human propensity to create mental models of their experiences. We construct, but do not discover, mathematics. Embodied mind theorists explain the effectiveness of mathematics by arguing that mathematics was constructed by the brain in order to be effective in this universe. It is part of our natural "model making" cognitive functioning. With this view, the physical universe can thus be seen as the ultimate foundation of mathematics: it guided the evolution of the brain and later determined which questions this brain would find worthy of investigation.
Regarding "laws of nature", I group them together with mathematical and logical objects. The "laws of nature" and do not cause the universe to be as it is. The universe is already as it is, and the laws are our human attempt to organize these experiences in ways that we can describe them, explain them, and predict them. An apple doesn't fall from the tree because Newton came up with the formulas for gravitational attraction, the formulas describe (and predict) how objects fall, and how they will fall in the future.
I am not a reductionist, in the sense that I don't think all behaviors can ultimately be reduced to their lowest common denominator (physical laws). The ultimate in reductionism would be to claim that quantum theory should eventually be able to explain not only how subatomic particles behave, but how atoms and molecules form, now cells come into being, now animals work, and finally how humans and human societies work. No, I am an "emergentist". I think behaviors spontaneously emerge as complexity increases. The characteristics of water (its wetness, its freezing point, boiling point, and chemical properties) spontaneously emerge as water molecules form from hydrogen and oxygen. We cannot "average" the characteristics of two hydrogen atoms and one oxygen atom to predicts what their chemical combination (H2O) will be like. The properties of water emerge as water (a more complex thing) is built from oxygen and hydrogen (less complex things). As molecules form, organic compounds are created, life emerges, intelligence evolves, and civilizations are born and die, new properties and behaviors spring into existence with each change in structure and complexity. The ultimate in complex properties - consciousness, intelligence, and human culture, exist in ourselves and our civilizations.